Conflict-free chromatic index of trees

Abstract

A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color that is assigned to exactly one edge among the closed neighborhood of $e$. The smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the conflict-free chromatic index of $G$, denoted $χ'_{CF}(G)$. Dȩbski and Przyby\a{l}o showed that $2\leχ'_{CF}(T)\le 3$ for every tree $T$ of size at least two. In this paper, we present an algorithm to determine the conflict-free chromatic index of a tree without 2-degree vertices, in time $O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and Przyby\a{l}o.

Figures (7)

• Figure 1: $S$-vertex and $D$-vertex (red is the conflict-free color of $T$)
• Figure 2: Coloring patterns for maximal full subtrees
• Figure 3: The sum of some trees
• Figure 4: Partial edge-colorings of $\mathcal{F}_1,\mathcal{F}_2,\mathcal{F}_3,\mathcal{F}_4$
• Figure 5: Partial edge-colorings of $\mathcal{F}_2$ (extended)

Theorems & Definitions (15)

• Theorem 1.1: MJ2022KMP
• Lemma 2.2
• Theorem 2.3
• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• Proposition 3.4
• Theorem 3.5
• Corollary 3.6
• Theorem 3.7